On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra

Perez-Sanchez, Carlos I.

doi:10.48550/arxiv.2007.10914

Cited by 4 publications

(9 citation statements)

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“…Note that the finite size of N might strongly affect these values. Another possibility is through functional renormalization group techniques [37]. We hope to explore these ideas in future works.…”

Section: Discussionmentioning

confidence: 99%

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Hessam

Khalkhali

Nathan

2023

J. Phys. A: Math. Theor.

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In this paper we study ensembles of finite real spectral triples equipped with a path integralover the space of possible Dirac operators. In the noncommutative geometric setting of spectraltriples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus,this path integral serves as a noncommutative analogue of integration over metrics, a key featureof a theory of quantum gravity. From these integrals in the so-called double scaling limit we derivecritical exponents of minimal models from Liouville conformal field theory coupled with gravity.Additionally, the asymptotics of the partition function of these models satisfy differential equationssuch as Painlev´e I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory.This is all proven using well-established and rigorous techniques from random matrix theory.

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Section: Discussionmentioning

confidence: 99%

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Hessam

Khalkhali

Nathan

2023

J. Phys. A: Math. Theor.

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“…Note that the finite size of N might strongly affect these values. Another possibility is through functional renormalization group techniques [36]. We hope to explore these ideas in future works.…”

Section: Discussionmentioning

confidence: 99%

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Hessam¹,

Khalkhali²,

Nathan³

2022

Preprint

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In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.

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“…In order to reach a continuum limit resembling smooth spin manifolds, the Functional Renormalization Group could be helpful in searching the fixed points (cf. the companion paper [Pér21] for the application of this idea to general multimatrix models).…”

Section: Discussionmentioning

confidence: 99%

“…Additional to such large-N one might require to adjust the couplings to criticality [BG16,Gla17,KP21]. This can also be addressed using the Functional Renormalization Group to find candidates for phase transition; for models still without matter, see [Pér21].…”

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confidence: 99%

“…As outlook (dashed), to reach a smooth geometry at the point S one needs a sensible limit (e.g. large-N and possibly tuning some parameters to criticality) in order to achieve phase transition part of this companion paper, the Functional Renormalization Group to multimatrix models [Pér21] inspired by random noncommutative geometry was addressed for some two-dimensional models obtained in [Pér19]. Solution of the matrix-models corresponding to one-dimensional geometries was addressed in [AK19], using Topological Recursion [EO07] (due to the presence of multitraces, in its blobbed [Bor15] version).…”

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confidence: 99%

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On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

Perez-Sanchez

2021

Preprint

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We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv:2007:10914, Ann. Henri Poincaré, 2021 we propose a gauge theory setting based on noncommutative geometry, which-just as the traditional formulation in terms of almost-commutative manifolds-has the ability to also accommodate a Higgs field. However, in contrast to 'almost-commutative manifolds', the present framework employs only finite dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang-Mills-Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here.

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On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra

Cited by 4 publications

References 42 publications

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Double scaling limits of Dirac ensembles and Liouville quantum gravity

On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

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